Interval vs Ratio Scales Notes

Interval and Ratio Scales

The transcript fragment indicates a distinction: when you have an interval variable, you cannot rely on ratios as meaningful comparisons. Ratios are meaningful on ratio scales but not on interval scales.
This leads to the key idea: different measurement scales support different kinds of arithmetic and interpretation.

Nominal scale: categories without order (e.g., blood type A, B, O).
Ordinal scale: ordered categories, with meaningful order but not equal intervals (e.g., Likert-scale ratings).
Interval scale: numeric scale with equal intervals, but lacking a true zero point (e.g., Celsius temperatures).
Ratio scale: numeric scale with equal intervals and a true zero (e.g., height, weight, kelvin temperature).

For interval data, you can compare differences, not ratios.
- You can say two subjects differ by a fixed amount, but you cannot claim that one value is a multiple of another.
- Example intuition: if subject A has 20 units and subject B has 10 units on an interval scale, you can say A is 10 units higher than B, but you cannot say A is twice B.
Formal implication: multiplicative comparisons (ratios) are not meaningful on interval scales because the zero point is arbitrary.

The difference is meaningful: difference between two measurements is interpretable as a change of one unit.
The order is meaningful: if x1 > x2, then subject 1 has a higher score/value than subject 2.
Ratios are not meaningful:
- You should not interpret x1/x2 as having a real, unit-consistent meaning when the scale is interval.

Celsius temperature (°C) as an interval variable:
- Equal intervals: a change from 20° to 30° is a 10-degree difference, just as from 5° to 15°.
- Zero point is arbitrary: 0°C does not mean 'no temperature'.
- Ratios are not meaningful: 20°C is not twice as hot as 10°C.
Kelvin temperature (K) as a ratio variable:
- Kelvin has an absolute zero; ratios are meaningful (e.g., 300 K is twice 150 K).
Height in centimeters (cm) or mass in kilograms (kg) as ratio variables:
- True zero exists; ratios are meaningful (e.g., 180 cm is twice 90 cm).

For interval and ratio data, you can compute:
- Mean: ar{x} = \frac{1}{n} \sum{i=1}^{n} xi
- Standard deviation: s = \sqrt{\frac{1}{n-1} \sum{i=1}^{n} (xi - \bar{x})^2}
- Variance: s^2 = \frac{1}{n-1} \sum{i=1}^{n} (xi - \bar{x})^2
For ordinal data, you generally should not rely on means or standard deviations; use medians, order-based methods, and nonparametric tests.

Sample mean: \bar{x} = \frac{1}{n} \sum{i=1}^{n} xi
Sample variance: s^2 = \frac{1}{n-1} \sum{i=1}^{n} (xi - \bar{x})^2
Sample standard deviation: s = \sqrt{\frac{1}{n-1} \sum{i=1}^{n} (xi - \bar{x})^2}
Difference between two interval measurements: \Delta = x1 - x2 (meaningful as a change, not as a ratio)
Ratio interpretation is meaningful for ratio scales but not for interval scales (e.g., if y is on a ratio scale, y1/y2 has interpretable meaning under a true zero).

Before performing calculations, identify the measurement scale of each variable.
Apply appropriate statistics:
- Interval/ratio: use means, standard deviations, t-tests, ANOVA, regression, etc.
- Ordinal: prefer medians, nonparametric tests (e.g., Mann-Whitney, Kruskal-Wallis).
When describing differences for interval data, report differences and confidence intervals for means, not ratios.
Be mindful of zero interpretation:
- Arbitrary zero on interval scales prevents meaningful ratio statements.
- True-zero scales (ratio) permit multiplicative comparisons.

Measurement theory: the nature of the scale determines permissible mathematical operations.
Derivation of statistical methods depends on scale properties (e.g., parametric tests assume interval/ratio data).
Conceptual clarity avoids misinterpretation of results (e.g., claiming a variable is 'twice as large' on an interval scale).

Misusing ratio-based language on interval data can mislead interpretations and decision-making.
Clear reporting of scale type improves transparency and reproducibility.
In practice, transform data cautiously when needed (e.g., converting to a ratio scale where appropriate, or using nonparametric methods when scale assumptions are violated).

The provided fragment begins with: "Ratios are not. K? So if you have an interval variable, you could say that two subjects are …" which aligns with the distinction discussed above: on interval data, you can discuss differences between subjects, not their ratios.
If you provide the remaining portion of the transcript, I can tailor these notes to mirror the exact wording and include any additional examples or nuances mentioned.